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Abstract

We conduct a detailed study of the existence theory for nonlinear hemivariational inequalities of second order. The problems under consideration are strongly nonlinear and not necessarily of variational nature. So we employ a variety of tools in order to solve them. More precisely, we use the general theory of nonlinear operators of monotone type, the method of upper-lower solutions, the multivalued Leray–Schauder principle, nonsmooth critical point theory coupled with Landesman–Lazer conditions and linking techniques and also truncation and penalization techniques. The problems that we examine involve Dirichlet boundary conditions; in the last section we also examine a problem with a nonhomogeneous and nonlinear Neumann boundary condition.